Astron. Astrophys. 332, 1099-1122 (1998)

6. Model calculation for a protoplanetary disk

6.1. Equations for the disk structure

The calculation of the structure of a protoplanetary accretion disk is based on the semi-analytical model for a thin stationary accretion disk of Duschl et al. (1996) ⁴. The details of the approximations on which this model is based on are described there. The resulting basic equations for the disk structure are

where s is the radial distance from the protosun in units AU, is the surface density, h the (half) thickness of the disk, M the mass of the protostar in solar masses, the constant accretion rate in units of , and P and T are the pressure and temperature in the midplane of the disk, respectively. is the Rosseland mean mass extinction coefficient and µ the mean molecular weight. The opacity is determined by the opacity of the dust and the gas. The approximation for due to dust used in the model calculation is described below. The opacity of the gas component is calculated from the analytical approximations given by Bell & Lin (1994). The details of the approximation of the gas opacity as used in this model calculation are described in Finocchi et al. (1997). The basic parameters of the disk used for the model calculation are listed in Table 2.

Table 2. Model parameters of the accretion disk used in the computation of disk structure

The inwards directed drift velocity of the disk material is

The characteristic timescale for a significant change in temperature follows from (63) and (65) as

(see Fig. 1). This corresponds to the characteristic timescale for temperature changes experienced by a dust grain as it spirals inwards with a radial drift velocity component given by (65).

6.2. Absorption by dust grains

The disk structure depends strongly on the mass extinction coefficient of the disk material. Throughout most part of the accretion disk the opacity is dominated by the strong absorption and scattering by the dust grains. As we have seen above, there exist many different kinds of dust species in different parts of the accretion disk, depending on the local pressure and temperature conditions. At least the contribution of the most abundant of the various species to the total opacity has to be considered in a model calculation of the disk structure. The dust mixture used in this calculation is that of the P94 model of Pollack et al. (1994) as specified in Sect.  2and the mixture of aluminium compounds predicted from chemical equilibrium calculations at temperatures where silicate and iron grains are already destroyed.

The composition of the P94 dust mixture does not correspond to the mixture obtained by any thermodynamic equilibrium composition at some pressure P and temperature T. It results from contributions from various sources, stellar and interstellar ones, where it is believed to be formed under extreme non-equilibrium conditions. The mixture will be subject to changes in the relative fraction of its constituents as the inwards drifting disk material gradually becomes warmer. This has to be considered in calculating the dust opacity.

In calculating the opacity of the dust we do not consider the full set of dust components existing in the disk but only the following four species:

1. The astronomical silicate as defined in the model of Draine & Lee (1984). The complex index of refraction given by Draine (1985) is used for calculating the extinction by the silicate dust components of the P94 dust mixture in the cold parts of the accretion disk ( K) where annealing of the disordered lattice structure of interstellar dust is inefficient. We prefer this over the silicate absorption data used by Pollak et al. (1994) since in the mid infrared the data of Draine seem to give the best representation of dust extinction (e.g. Mathis 1996, Li & Greenberg 1997) and it is the opacity in this spectral region that determines the disk structure between the region of ice vapourisation and annealing of the amorphous silicate dust.
2. Olivine as a model for a silicate with a well ordered lattice structure. We do not discriminate in our extinction calculation between the two abundant silicate species olivine and orthopyroxene since both have a rather similar complex dielectric coefficient (cf. Fig. 1 of Pollack et al. 1994for instance) and take olivine as being representative for both materials. The olivine is used for calculating the extinction by silicates after annealing took place. Data for the complex index of refraction of crystalline olivine in the wavelength region 80 Å m have been obtained several years ago from D.R. Huffman (private communication, c.f. also Huffman & Stapp 1973)). The crystalline olivine shows a much smaller absorption than the "dirty astronomical silicate" of Draine (1984) or the glassy olivine studied by Dorschner et al. (1995).
3. Iron metal. For the wavelength region m we use data for the complex index of refraction as given in the CRC handbook (Lide 1995). For longer wavelength we use the same data as Pollak et al. (1994).
4. Corundum is chosen as a representative absorber for the aluminium compounds. We do not discriminate in our calculation of the extinction between the different aluminium dust species but treat them all as being corundum. Data for the complex index of refraction of corundum are taken from Koike et al. (1995). We use their ISAS-data representing the absorption properties of fine corundum grains produced by combustion of solid-rocket propellants. We believe that the formation conditions of such grains are roughly comparable to a corundum condensate formed in circumstellar shells.

In our present model calculation we do not consider the carbon dust component since we have not included the complex chemistry of carbon dust destruction (see Finocchi et al. 1997) in this model calculation. The troilite dust component of the P94 dust mixture is neglected in our calculation of the dust opacity, since this species is only a minor absorber (eg. Pollack et al. 1994). It is simply treated as being iron in the opacity calculation. The SiO₂ dust component assumed to be present in the P94 dust mixture is treated in the opacity calculation as being a silicate.

Real iron grains in an accretion disk are not really pure iron particles but their material is a solid solution of iron with an admixture of several percent of nickel and small contents of some other elements. This is not considered in our calculations of the opacity of such grains. Further we do not consider that after annealing of the dirty astronomical silicate the crystalline silicate grains resulting from this may contain tiny inclusions of for instance iron grains or of aluminium-calcium compounds.

The Rosseland mean opacities of these four dust materials was calculated as follows: For a dense grid of particle radii a and frequencies the extinction coefficient was calculated by Mie theory for spherical grains for each a and . The complex index of refraction for the dust materials was chosen as discussed above. The reciprocal 's first are integrated numerically over the paricle size distribution and the result is used to calculate the Rosseland mean opacities by a numerical integration according to the standard definition of . The results are shown in Fig. 14. We assumed in this calculation that the dust grains are distributed in size between and m according to the widely accepted Mathis-Rumpl-Nordsiek (MRN) size distribution (Mathis et al. 1977). This is a reasonable assumption for the silicate dust grains since the MRN radius spectrum is assumed to describe well the size of interstellar grains. Though one expects that grains in the cold outer part of the accretion disk have agglomerated into bigger sized clusters of particles (Stognienko et al. 1995), these agglomerates are bound only by weak van der Waals forces. They are expected to disintegrate again in the warm parts of the accretion disk (T at least several 100 K), in which we are interested in this paper. More recent models for interstellar dust absorption favour a different distribution of grain sizes for the silicate grain component of the dust mixture (Li & Greenberg 1997) or for all components (Mathis 1996). Since the dust grains prior to the onset of planetesimal formation all are small compared to the wavelengths of interest even in the warm parts of the accretion disk, the extinction does not depend on the special choice of the size distribution of grains. In the cold part of the accretion disk, however, the grains are likely to have a porous fluffy structure and the extinction then depends on the size and structure of the dust aggregates (Miyake & Nakagawa 1993, Stognienko et al. 1995).

Fig. 14a-d. Rosseland mean of the mass extinction coefficient (in units cm2 / g) averaged with a MRN size distribution for some important dust species

For the iron and the aluminium dust component the assumption of a MRN distribution is rather arbitrary. It is used in default of any information on the real sizes of such grains.

As can be seen from Fig. 14 the most efficient absorber in the inner parts of the accretion disk is the "dirty astronomical silicate". It dominates the opacity of the disk matter until it is converted into crystalline silicate due to annealing at elevated temperatures (see below). After annealing of the amorphous silicate dust the opacity is dominated by the opacity of the iron grains which are more efficient absorbers than the crystalline silicates. Once the iron grains are vapourised the extinction is dominated by the aluminium compounds until also these grains are destroyed close to the star. The disappearance of certain dust materials with increasing temperature obviously is accompanied at each step by a strong decrease of the extinction coefficient. This has strong implications for the structure of the accretion disk, as we shall see in Sect.  8.

© European Southern Observatory (ESO) 1998

Online publication: March 30, 1998
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